Endpoint Boundedness of Riesz Transforms on Hardy Spaces Associated with Operators

TL;DR

This paper establishes the endpoint boundedness of Riesz transforms associated with certain operators on Hardy spaces, extending known results to the critical case where p equals n/(n+1).

Contribution

It proves the boundedness of Riesz transforms from Hardy spaces linked to operators to weak Hardy spaces at the critical exponent p=n/(n+1), filling a gap in the theory.

Findings

Riesz transforms are bounded from $H^p_{L_i}$ to $WH^p$ at the critical p.

Extension of boundedness results to the endpoint case.

Applicable to operators satisfying Davies-Gaffney estimates and elliptic operators with complex coefficients.

Abstract

Let $L_1$ be a nonnegative self-adjoint operator in $L^2({\mathbb R}^n)$ satisfying the Davies-Gaffney estimates and $L_2$ a second order divergence form elliptic operator with complex bounded measurable coefficients. A typical example of $L_1$ is the Schr\"odinger operator $-\Delta+V$, where $\Delta$ is the Laplace operator on ${\mathbb R}^n$ and $0\le V\in L^1_{\mathop\mathrm{loc}} ({\mathbb R}^n)$. Let $H^p_{L_i}(\mathbb{R}^n)$ be the Hardy space associated to $L_i$ for $i\in\{1,\,2\}$. In this paper, the authors prove that the Riesz transform $D (L_i^{-1/2})$ is bounded from $H^p_{L_i}(\mathbb{R}^n)$ to the classical weak Hardy space $WH^p(\mathbb{R}^n)$ in the critical case that $p=n/(n+1)$. Recall that it is known that $D (L_i^{-1/2})$ is bounded from $H^p_{L_i}(\mathbb{R}^n)$ to the classical Hardy space $H^p(\mathbb{R}^n)$ when $p\in(n/(n+1),\,1]$.